4.7 Parametrization of Implicitly Defined Functions

Introduction to Parametrization

Imagine you are watching a race car driving around a circular track. If you only have the equation of the circle, you know the shape of the track, but you don't know where the car is at any specific time or which direction it is moving. That is where parametrization comes in! In this chapter, we will learn how to take "implicitly defined" equations (where \(x\) and \(y\) are tangled together) and rewrite them using a third variable, usually \(t\) for time, called a parameter. This gives us a "play-by-play" description of the curve.

Note: This topic belongs to Unit 4, which is not assessed on the AP Precalculus Exam, but it is incredibly useful for future Calculus courses and physics!

What Does "Parametrizing" Mean?

In Topic 4.5, we looked at implicitly defined functions like \(x^2 + y^2 = 25\). In these equations, \(y\) isn't isolated. To parametrize an equation like this, we create two separate functions: one for \(x\) and one for \(y\), both depending on the same variable \(t\):

\(x = f(t)\)

\(y = g(t)\)

When you plug these "parametric equations" back into the original implicit equation, the equation must remain true. Think of \(t\) as the "invisible hand" that moves the point \((x, y)\) along the path.

Scenario 1: Parametrizing a Basic Function

If you already have a function in the form \(y = f(x)\), parametrizing it is very straightforward. Don't worry if this seems too simple—sometimes the simplest way is the best way!

The "Trivial" Parametrization:

1. Let \(x = t\)

2. Substitute \(t\) into the equation for \(x\) to get \(y = f(t)\)

Example: Parametrize the curve \(y = 3x^2 - 5\).

Step 1: Let \(x = t\)

Step 2: Since \(y = 3x^2 - 5\), then \(y = 3t^2 - 5\)

Our parametric equations are: \(\begin{cases} x = t \\ y = 3t^2 - 5 \end{cases}\)

Scenario 2: Parametrizing Circles (The Trig Trick)

For shapes like circles and ellipses (covered in Topic 4.6), the simple "let \(x = t\)" trick doesn't work well because these shapes aren't functions—they fail the vertical line test. Instead, we use our best friends from Unit 3: Sine and Cosine.

Recall the Pythagorean Identity: \(\cos^2(t) + \sin^2(t) = 1\).

To parametrize the circle \(x^2 + y^2 = r^2\):

We want \((\frac{x}{r})^2 + (\frac{y}{r})^2 = 1\). If we let \(\frac{x}{r} = \cos(t)\) and \(\frac{y}{r} = \sin(t)\), we get:

\(x = r \cos(t)\)

\(y = r \sin(t)\)

Usually, we let \(0 \leq t \leq 2\pi\) to trace the circle exactly once.

Example: Parametrize the circle \(x^2 + y^2 = 16\).

Here, the radius \(r\) is \(\sqrt{16} = 4\).
So, the equations are: \(\begin{cases} x = 4\cos(t) \\ y = 4\sin(t) \end{cases}\)

Scenario 3: Parametrizing Ellipses

An ellipse is just a "stretched" circle. The standard implicit form is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

To parametrize this, we assign the \(x\) and \(y\) parts to cosine and sine:

\(x = a \cos(t)\)

\(y = b \sin(t)\)

Quick Review: If you have \(\frac{x^2}{9} + \frac{y^2}{25} = 1\), then \(a = 3\) and \(b = 5\).
Your parametrization is: \(\begin{cases} x = 3\cos(t) \\ y = 5\sin(t) \end{cases}\)

Did you know?

Computer animators use parametrization to move characters. Instead of calculating the whole shape of a character's path at once, they use \(t\) (time) to determine exactly where a character's hand or foot should be at frame 60 versus frame 120!

Scenario 4: Parametrizing Lines

While we can use the "let \(x = t\)" method for lines, we often want to parametrize a line based on a starting point \((x_0, y_0)\) and a rate of change.

The parametric form of a line is:
\(x = x_0 + at\)
\(y = y_0 + bt\)

In this form, \((x_0, y_0)\) is where the object is when \(t = 0\), and the values \(a\) and \(b\) tell you how fast the \(x\) and \(y\) coordinates are changing per unit of time.

Example: Find parametric equations for a line passing through \((2, 5)\) with a slope of \(m = \frac{3}{4}\).

Slope is \(\frac{\text{change in } y}{\text{change in } x}\). So, we can say for every \(4\) units \(x\) moves, \(y\) moves \(3\) units.

\(x = 2 + 4t\)

\(y = 5 + 3t\)

Common Mistakes to Avoid

• Ignoring the Domain of \(t\): If you are parametrizing a line segment (a piece of a line), you must specify the interval for \(t\). For example, \(0 \leq t \leq 1\).
• Swapping Sine and Cosine: While \(x = r \sin(t)\) and \(y = r \cos(t)\) will still draw a circle, it changes the starting point and the direction (orientation) of the curve. By convention, we use \(x = \cos(t)\) to start on the positive x-axis and move counter-clockwise.
• Radius vs. Square of Radius: Remember to take the square root of the constant in a circle's equation (like \(x^2 + y^2 = 49\)) to find the \(r\) value (which is \(7\)) for your parametrization.

Summary and Key Takeaways

Key Takeaway 1: Parametrization breaks a single equation involving \(x\) and \(y\) into two separate equations based on a parameter \(t\).

Key Takeaway 2: For functions, use \(x = t\) and \(y = f(t)\).

Key Takeaway 3: For circles and ellipses, use the trigonometric identities: \(x = a \cos(t)\) and \(y = b \sin(t)\).

Key Takeaway 4: Parametric equations are not unique! There are many different ways to parametrize the same curve, depending on how fast you want to move along the path or where you want to start.